{"id":"acdfe1ec-c4e2-4dfa-8d94-040682ec9f01","arxiv_id":"2507.07782","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper defines nonlinear induced topological pressure and proves it equals the supremum of (metric entropy plus nonlinear potential) divided by the induced-time integral, over all invariant measures.","lead":"This paper introduces a high-dimensional nonlinear version of induced topological pressure, a complexity measure for dynamical systems with a time-changing clock, and proves a variational principle for it. It also works out properties of the classical induced pressure, including equilibrium states and freezing states.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified: Theorem 4.6 is logically sound under its stated hypotheses; the abundance condition is strong but explicit and non-vacuous, so the main gap is expository.","rationale":"The central claim is Theorem 4.6. To sustain it one needs (i) the root characterization P^F_ψ(Φ)=inf{β:P^{Gβ}(Ψ)≤0} from Corollary 4.2, and (ii) the nonlinear variational formula for P^{Gβ} from [3]. Checking the proof of Corollary 4.2, the strict monotonicity argument is correct: for β1<β2 and 0<ε≤m(β2-β1)/2, the approximate maximizer for β2 gives P^{Gβ2}<P^{Gβ1}, and the root equality follows from continuity and the strict decrease. The proof of Theorem 4.5, the technical heart of the reduction, is also sound: the lower bound uses disjoint hitting times T_j with T_{j+1}-T_j>2||ψ|| to force an infinite exponential series; the upper bound groups separated sets by the unique ℓ with (ℓ-1)m<S_nψ≤ℓm and then uses the growth bound on P^F_{ψ,ℓm}. The undefined m is an exposition slip and should be read as m=inf ψ. Theorem 4.6's proof contains a sign typo in the first branch, where '0 >' should be '0 ≥' or simply '≤', but the inequality actually needed is (hν+F(∫Φdν))/∫ψ dν ≤ β for all β>P, which yields the desired lower bound on P^F_ψ(Φ). The abundance condition is strong but explicit; it is not used circularly, and it is known to hold for systems with entropy-dense ergodic measures, such as full shifts. Thus the paper's main weakness is scope and illustration, not correctness of the central argument.","tokens_in":24201,"tokens_out":26179,"duration_ms":307415,"concrete_test":"Check the non-vacuity of Definition 4.1 for a full shift on {0,1} with any continuous Ψ: use entropy density of ergodic measures to produce, for each μ, h<hμ( f), and ε>0, an ergodic ν with hν( f)>h and |∫Ψ dν - ∫Ψ dμ|<ε. If this check succeeds, the nonconvex branch of Theorem 4.6 applies to a concrete nonconvex F (e.g., F(a)=a^3), and the remaining concern is expository rather than mathematical.","verdict_should_be":"UNCHANGED","load_bearing_attack":"I find no load-bearing flaw in the central variational principle. Theorem 4.6 follows from Corollary 4.2 plus the cited nonlinear variational principle: for β below the root, P^{Gβ}(Ψ)>0 gives the upper bound by choosing a measure with hν+F(∫Φdν)-β∫ψ dν≥0, and for β above the root, P^{Gβ}(Ψ)<0 gives the lower bound pointwise. The typographic '0 >' in the first half of the proof should read '0 ≥' (or '≤'), but the intended conclusion is unaffected. The abundance hypothesis in Definition 4.1 is indeed strong, but it is an explicit hypothesis, not a hidden one; the paper would be easier to use with an example, but its absence does not invalidate the theorem. The convex branch is a direct application of the cited theorem from [3]. Minor issues in Theorem 2.3(vi), the undefined m in Theorem 4.5, and the sign typo in Theorem 4.6 are presentational.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies induced topological pressure in two settings. For the classical induced pressure introduced by Xing and Chen, it provides equivalent open-cover definitions and elementary properties such as monotonicity, continuity, convexity, bounds, and a characterization of invariant measures. It then develops equilibrium states, subdifferentials, and freezing/zero-temperature states for this pressure, extending earlier work of Hedges. The second half introduces a high-dimensional nonlinear induced topological pressure, proves a root representation in terms of the nonlinear pressure of an augmented potential (Theorem 4.5), and establishes a variational principle (Theorem 4.6) under either an abundance-of-ergodic-measures condition or convexity of the nonlinearity F. The main result unifies the classical induced-pressure variational principle and the higher-dimensional nonlinear pressure of Barreira and Holanda.","tokens_in":24309,"tokens_out":31300,"duration_ms":335710,"significance":"If the results are correct, the paper gives a useful unified formalism: the nonlinear induced pressure is shown to equal the supremum of (hν + F(∫Φ dν))/∫ψ dν over invariant measures, with the root characterization of Theorem 4.5 as an intermediate tool. The proofs are honest and transparent: the central derivation relies on the cited variational principles of Xing-Chen and Barreira-Holanda, and I found no circularity or parameter-fitting. The abundance condition in Definition 4.1 is indeed strong, but it is an explicit hypothesis rather than a hidden assumption; the paper would be easier to use with examples, and I do not regard the absence of examples as a correctness defect. The main chain from Theorem 4.5 through Corollary 4.2 to Theorem 4.6 is logically coherent, and the secondary material on equilibrium states and freezing states is mostly sound, with one local proof gap noted below.","major_comments":[],"minor_comments":[{"comment":"The symbol m is used throughout the proof, for instance in the intervals (ℓ−1)m < Snψ(x) ≤ ℓm, but it is never defined in the statement. From the context it must be m = inf ψ; it should be introduced explicitly before the proof.","section":"Theorem 4.5"},{"comment":"In the upper-bound half, the displayed line '0 > (hν(f)+F(∫Φdν))/∫ψdν − β' should read '0 ≥ ...'. The intended conclusion is unaffected, but as written the inequality does not follow from the preceding line P^{Gβ}(Ψ) ≤ 0.","section":"Theorem 4.6 proof"},{"comment":"The proof invokes Theorem 3.3, but Theorem 3.3 requires a subset F to be the full set of equilibrium states and requires property (2) to hold for all members of F. The present proof does not show that {μ} is the entire equilibrium set, nor that every equilibrium state maximizes ∫ϕ/∫ψ. The corollary is nevertheless true by a direct argument using hν+∫ϕ/∫ψ ≤ Pψ(ϕ) for all ν, so the proof should be rewritten accordingly.","section":"Corollary 3.2"},{"comment":"The continuity of the map β ↦ P^{Gβ}(Ψ), which is needed for the equality inf{β : P^{Gβ}(Ψ) ≤ 0} = sup{β : P^{Gβ}(Ψ) ≥ 0}, is asserted without proof. It follows from the continuity of nonlinear pressure in the potential, but the relevant estimate should be stated explicitly.","section":"Corollary 4.2"},{"comment":"The abundance condition is strong and no concrete system satisfying it is mentioned. A remark giving standard examples, such as mixing subshifts of finite type where ergodic measures are entropy dense, would significantly improve the paper's usability and make clear that the hypothesis is non-vacuous.","section":"Definition 4.1 and Theorem 4.6"},{"comment":"There are several small typographical and notational issues: in Theorem 4.4 the set SY_T is defined with 'y ∈ X' instead of 'y ∈ Y'; Corollary 4.1 writes lim_{T→∞} where limsup_{T→∞} is meant; Corollary 3.1 contains 'Morevoer'; the proof of Theorem 4.3 uses the notation 'P ψ F' in place of 'P^F_ψ'; and Remark 4.2 delegates an open-cover equivalence to 'similarly' without a sketch. These should be corrected in a final pass.","section":"Throughout"}],"recommendation":"minor_revision","confidential_remarks":"The manuscript is a competent extension of [28] and [3]. I agree with the stress-test assessment: the concern about the abundance hypothesis is not a correctness issue, since it is explicitly stated and standard systems satisfy it. The main variational principle is sound, and the remaining problems are local. I would not require a fresh round of external review after the requested revisions."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThe paper does one genuinely new thing: it defines a high-dimensional nonlinear induced topological pressure (Section 4.2) and proves a variational principle for it (Theorem 4.6). That result is not in the cited work of Xing–Chen, Barreira–Holanda, or Hedges. The proof strategy is sensible: relate the new pressure to the root of an ordinary nonlinear pressure (Corollary 4.2), then invoke the variational principle from Barreira–Holanda. I checked the main line and it holds. The earlier sections, on equivalent definitions, equilibrium states, and freezing states for the classical induced pressure, are mostly routine extensions of existing arguments. They are competently done.\n\nThe weakest point is the abundance hypothesis in Definition 4.1. The paper gives no example of a system satisfying it, and it is a nontrivial condition on the pair (f, Ψ). The convexity branch of Theorem 4.6 avoids it, so the result is not hostage to an unverifiable assumption. But the strongest form of the theorem is conditional on a hypothesis the authors never unpack. That is a real presentational gap, not a mathematical error.\n\nMinor issues: an undefined 'm' in the proof of Theorem 4.5, and a reversed inequality sign in the middle of the proof of Theorem 4.6 (the '0 >' should be '0 ≥' or '≤'; the conclusion is unaffected). The reader's report flagged a flawed spanning-set argument in Theorem 2.3(vi). I do not see a real flaw there — for separated sets the inequality Σ a_x b_x ≤ (Σ a_x)(Σ b_x) gives P(φ+g) ≤ P(φ)+P(g) directly. The text uses spanning sets and writes the product bound for sums over n, which is slightly sloppy, but the statement is true and the proof is repairable.\n\nThe citation practice is honest. The new part builds explicitly on [3] and [28], and self-citation is not an issue. There is no fitting of data or circular reasoning.\n\nBottom line: this is a solid, incremental paper. It will be useful to people working in thermodynamic formalism, multifractal analysis, and large deviations. It deserves a serious referee. I would send it out, with the expectation of minor revision on exposition and hopefully an example for the abundance condition.","headline":"Competent extension of induced pressure to the nonlinear setting; the new variational principle is real and the proof holds, but the main theorem is conditional on a strong hypothesis the paper never illustrates.","tokens_in":24918,"tokens_out":5589,"would_cite":false,"duration_ms":57393,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["37D35","28D20","37A35"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that nonlinear induced topological pressure satisfies a variational principle relating it to entropy and potential integrals over invariant measures.","keywords":["induced topological pressure","variational principle","nonlinear thermodynamic formalism","equilibrium state","freezing state","subdifferential","topological entropy","ergodic measures"],"falsifier":"Take a non-convex $F$, for example $F(u,v)=-(u-v)^2$, on a system whose ergodic measures are too sparse to approximate the invariant measures entering the supremum; compute $P^F_\\psi(\\Phi)$ from the defining spanning-set limit and compare it with the right-hand side of Theorem 4.6. Any strict gap between the two numbers would refute the variational principle.","tokens_in":1860,"feed_emoji":"📐","tokens_out":1888,"duration_ms":115094,"temperature":0.7,"pith_summary":"This paper introduces a high-dimensional nonlinear version of induced topological pressure, $P^F_\\psi(\\Phi)$, which counts orbit complexity at time scales selected by a positive continuous function $\\psi$ while weighting a vector of observables $\\Phi$ through a continuous function $F$. The main result is a variational principle: if either $F$ is convex or the augmented system $(f,(\\Phi,\\psi))$ has an abundance of ergodic measures, then $P^F_\\psi(\\Phi)=\\sup_{\\nu\\in M(X,f)} (h_\\nu(f)+F(\\int\\Phi\\,d\\nu))/\\int\\psi\\,d\\nu$. For $d=1$ and $F=\\mathrm{id}$ this recovers the known induced-pressure variational principle, and with $\\psi\\equiv1$ it reduces to the classical topological-pressure variational principle. The paper also proves properties of the classical induced pressure, including equilibrium states, subdifferentials, and a freezing-state criterion.","feed_headline":"Nonlinear induced pressure equals a measure-theoretic supremum","feed_subtitle":"The new quantity equals the best entropy-plus-potential quotient over invariant measures.","key_machinery":"The load-bearing construction is the $\\psi$-induced partition of time: for each horizon $T$, the set $S_T=\\{n:\\exists x,\\ S_n\\psi(x)\\le T<S_{n+1}\\psi(x)\\}$ records the relevant return times, and $X_n=\\{x:S_n\\psi(x)\\le T<S_{n+1}\\psi(x)\\}$ are the corresponding orbit strips. The nonlinear induced pressure is built from these by summing $\\exp(nF(S_n\\Phi(x)/n))$ over $(n,\\varepsilon)$-spanning or separated sets inside each $X_n$ and taking $\\lim_{\\varepsilon\\to0}\\limsup_{T\\to\\infty}\\frac1T\\log(\\cdot)$. The proof's pivot is the augmented vector $\\Psi=(\\Phi,\\psi)$ together with the one-parameter family $G_\\beta(a,b)=F(a)-\\beta b$, since the induced pressure turns out to be the crossing point where the ordinary nonlinear pressure $P^{G_\\beta}(\\Psi)$ changes sign.","core_discovery":"The paper's central claim is Theorem 4.6. For a continuous map on a compact metric space, a positive continuous scaling function $\\psi$, a vector of potentials $\\Phi=(\\varphi_1,\\dots,\\varphi_d)$, and a continuous $F:\\mathbb{R}^d\\to\\mathbb{R}$, the high-dimensional nonlinear induced topological pressure satisfies $$P^F_\\psi(\\Phi)=\\sup_{\\nu\\in M(X,f)}\\frac{h_\\nu(f)+F(\\int\\Phi\\,d\\nu)}{\\int\\psi\\,d\\nu}$$ whenever either $F$ is convex or $(f,\\Psi)$ with $\\Psi=(\\Phi,\\psi)$ has an abundance of ergodic measures. The bridge is Corollary 4.2: with $G_\\beta(a,b)=F(a)-\\beta b$, the induced pressure is the common infimum and supremum of the set of $\\beta$ where the ordinary nonlinear pressure $P^{G_\\beta}(\\Psi)$ is non-positive or non-negative, so $P^F_\\psi(\\Phi)$ is the root of the map $\\beta\\mapsto P^{G_\\beta}(\\Psi)$. Feeding the known nonlinear variational principle for $P^{G_\\beta}$ into that root equation produces the quotient formula.","pith_inferences":["If the abundance condition is genuinely hard to verify, the convex branch is the operative version of Theorem 4.6, and concrete systems satisfying the non-convex branch would substantially widen its reach.","The root characterization suggests a practical numerical route: approximate $P^{G_\\beta}(\\Psi)$ for a range of $\\beta$ and bisect, which would compute induced pressure even when the variational formula is not covered by the theorem.","The freezing-state criterion of Section 3 should carry over to the nonlinear family $P_\\psi(\\beta\\Phi)$, since Theorem 4.6 makes that one-parameter family available and zero-temperature limits would then select maximizing measures of the normalized nonlinear potential.","Because the construction only requires a positive scaling function and pressure machinery, the same induced-pressure scheme may extend to flows or random dynamical systems, a direction the paper does not pursue."],"forward_implications":["For convex $F$, the variational principle holds for every topological dynamical system, so the induced pressure is obtained directly from invariant measures without extra dynamical hypotheses.","Through Corollary 4.2, the induced pressure can be located by finding the zero of $\\beta\\mapsto P^{G_\\beta}(\\Psi)$, reducing one new thermodynamic quantity to an existing one.","For $d=1$ and $F=\\mathrm{id}$, all the Section 2 and Section 3 results apply to the classical induced pressure $P_\\psi(\\varphi)$, including open-cover definitions, convexity, continuity, and cocycle invariance.","Equilibrium states of the induced pressure form a convex set whose extreme points are ergodic, and a potential freezes exactly when the pressure curve becomes affine with slope $\\mathrm{Max}_\\psi(\\varphi)$ and intercept $h^\\psi_\\infty(\\varphi)$ (Theorem 3.4).","The nonlinear induced pressure is invariant under topological conjugacy after pulling back $\\Phi$ and $\\psi$, matching the invariance pattern of the ordinary nonlinear pressure (Theorem 4.4)."],"supporting_citations":[{"why":"Defines the classical induced topological pressure for general topological dynamical systems and proves its variational principle, which this paper extends to the nonlinear setting.","marker":"[28]"},{"why":"Supplies the definition of higher-dimensional nonlinear topological pressure and the nonlinear variational principle used in Theorem 4.1 and Corollary 4.2.","marker":"[3]"},{"why":"Introduces induced topological pressure for Markov shifts with a scaling function, the origin of the $\\psi$-induced framework.","marker":"[14]"},{"why":"Provides the freezing-state and zero-temperature-limit notions that Section 3 adapts to induced pressure.","marker":"[13]"},{"why":"Develops nonlinear thermodynamical formalism for generalized mean-field models, the conceptual source of nonlinear pressure.","marker":"[8]"},{"why":"Supplies the standard entropy, topological pressure, and variational-principle background used throughout the paper.","marker":"[27]"}],"fun_headline_variants":["Induced pressure: quotient of entropy plus potential","Nonlinear induced pressure: max over invariant measures","High-dim induced pressure: a variational formula","Induced topological pressure: root of a pressure map"],"cache_read_input_tokens":27008,"weakest_assumption_plain":"The non-convex half of Theorem 4.6 rests on the 'abundance of ergodic measures' hypothesis for the augmented vector, and the paper gives no concrete system known to satisfy it; if that hypothesis fails and $F$ is not convex, the quotient formula has no proof and can fail.","fun_headline_variants_meta":{"raw":{"variants":["Induced pressure: quotient of entropy plus potential","Nonlinear induced pressure: max over invariant measures","High-dim induced pressure: a variational formula","Induced topological pressure: root of a pressure map"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001282,"raw_usage":{"total_tokens":5174,"prompt_tokens":817,"completion_tokens":4357,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":433,"completion_tokens_details":{"reasoning_tokens":4298}},"tokens_in":433,"tokens_out":4357,"duration_ms":28797,"temperature":1.0,"reasoning_tokens":4298,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T18:34:55.457968+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a non-convex $F$, for example $F(u,v)=-(u-v)^2$, on a system whose ergodic measures are too sparse to approximate the invariant measures entering the supremum; compute $P^F_\\psi(\\Phi)$ from the defining spanning-set limit and compare it with the right-hand side of Theorem 4.6. Any strict gap between the two numbers would refute the variational principle.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines the classical induced topological pressure for general topological dynamical systems and proves its variational principle, which this paper extends to the nonlinear setting."},{"cited_title":"Barreira, C","cited_arxiv_id":null,"evidence_quote":"Supplies the definition of higher-dimensional nonlinear topological pressure and the nonlinear variational principle used in Theorem 4.1 and Corollary 4.2."},{"cited_title":"Jaerisch, M","cited_arxiv_id":null,"evidence_quote":"Introduces induced topological pressure for Markov shifts with a scaling function, the origin of the $\\psi$-induced framework."},{"cited_title":"Buzzi, B","cited_arxiv_id":null,"evidence_quote":"Develops nonlinear thermodynamical formalism for generalized mean-field models, the conceptual source of nonlinear pressure."},{"cited_title":"Walters, An Introduction to Ergodic Theory, Graduate Texts in Mathematics, Springer- Verlag, 1982","cited_arxiv_id":null,"evidence_quote":"Supplies the standard entropy, topological pressure, and variational-principle background used throughout the paper."}],"review_version":1}